Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-06-28T14:39:40.008Z Has data issue: false hasContentIssue false
  • English
  • Français

Out-of-plane buckling in two-dimensional glass drawing

Published online by Cambridge University Press:  29 April 2019

D. O’Kiely Open the ORCID record for D. O’Kiely [Opens in a new window] ,
C. J. W. Breward ,
I. M. Griffiths Open the ORCID record for I. M. Griffiths [Opens in a new window] ,
P. D. Howell  and
U. Lange
D. O’Kiely*
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, OxfordOX2 6GG, UK
C. J. W. Breward
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, OxfordOX2 6GG, UK
I. M. Griffiths
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, OxfordOX2 6GG, UK
P. D. Howell
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, OxfordOX2 6GG, UK
U. Lange
Affiliation:
Schott AG, Hattenbergstrasse 10, 55122 Mainz, Germany
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We derive a mathematical model for the drawing of a two-dimensional thin sheet of viscous fluid in the direction of gravity. If the gravitational field is sufficiently strong, then a portion of the sheet experiences a compressive stress and is thus unstable to transverse buckling. We analyse the dependence of the instability and the subsequent evolution on the process parameters, and the mutual coupling between the weakly nonlinear buckling and the stress profile in the sheet. Over long time scales, the sheet centreline ultimately adopts a universal profile, with the bulk of the sheet under tension and a single large bulge caused by a small compressive region near the bottom, and we derive a canonical inner problem that describes this behaviour. The large-time analysis involves a logarithmic asymptotic expansion, and we devise a hybrid asymptotic–numerical scheme that effectively sums the logarithmic series.

JFM classification

Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 869 , 25 June 2019 , pp. 587 - 609
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Batty, C., Uribe, A., Audoly, B. & Grinspun, E. 2012 Discrete viscous sheets. ACM Trans. Graph. 31 (4), 113.Google Scholar
Buckmaster, J. D., Nachman, A. & Ting, L. 1975 The buckling and stretching of a viscida. J. Fluid Mech. 69 (01), 1–20.Google Scholar
Burke, S.2016 This glass can bend in half without shattering (online video clip). http://edition.cnn.com/videos/cnnmoney/2016/03/18/bendable-glass-schott-burke-pkg.cnn-money/video/playlists/technology/, CNN Money.Google Scholar
Chiu-Webster, S. & Lister, J. R. 2006 The fall of a viscous thread onto a moving surface: a ‘fluid-mechanical sewing machine’. J. Fluid Mech. 569, 89–111.Google Scholar
Cruickshank, J. O. 1988 Low-Reynolds-number instabilities in stagnating jet flows. J. Fluid Mech. 193, 111–127.Google Scholar
Cruickshank, J. O. & Munson, B. R. 1981 Viscous fluid buckling of plane and axisymmetric jets. J. Fluid Mech. 113, 221–239.Google Scholar
Davidovitch, B., Schroll, R. D., Vella, D., Adda-Bedia, M. & Cerda, E. A. 2011 Prototypical model for tensional wrinkling in thin sheets. Proc. Natl Acad. Sci. USA 108 (45), 18227–18232.Google Scholar
Filippov, A. & Zheng, Z. 2010 Dynamics and shape instability of thin viscous sheets. Phys. Fluids 22 (2), 023601.Google Scholar
Horvatitsch, T.2016 Ultra-thin glass. Schott Solutions 1/2016, pp. 6–11. Schott AG.Google Scholar
Howell, P. D.1994 Extensional thin layer flows. PhD thesis, University of Oxford.Google Scholar
Howell, P. D. 1996 Models for thin viscous sheets. Eur. J. Appl. Maths 7 (04), 321–343.Google Scholar
Kropinski, M. C. A., Ward, M. J. & Keller, J. B. 1995 A hybrid asymptotic–numerical method for low Reynolds number flows past a cylindrical body. SIAM J. Appl. Maths 55 (6), 1484–1510.Google Scholar
Mahadevan, L., Ryu, W. S. & Samuel, A. D. T. 1998 Fluid ‘rope trick’ investigated. Nature 392 (6672), 140–140.Google Scholar
Matovich, M. A. & Pearson, J. R. A. 1969 Spinning a molten threadline. Steady-state isothermal viscous flows. Ind. Engng Chem. Fundam. 8 (3), 512–520.Google Scholar
Morris, S. W., Dawes, J. H. P., Ribe, N. M. & Lister, J. R. 2008 Meandering instability of a viscous thread. Phys. Rev. E 77 (6), 066218.Google Scholar
O’Kiely, D.2018 Mathematical models for the glass sheet redraw process. PhD thesis, University of Oxford.Google Scholar
O’Kiely, D., Breward, C. J. W., Griffiths, I. M., Howell, P. D. & Lange, U. 2015 Edge behaviour in the glass sheet redraw process. J. Fluid Mech. 785, 248–269.Google Scholar
O’Kiely, D., Breward, C. J. W., Griffiths, I. M., Howell, P. D. & Lange, U. 2018 Glass sheet redraw through a long heater zone. IMA J. Appl. Maths 83 (5), 799–820.Google Scholar
Perdigou, C.2015 Stability of viscous sheets in open flow. PhD thesis, Université Pierre et Marie Curie-Paris VI.Google Scholar
Ribe, N. M. 2001 Bending and stretching of thin viscous sheets. J. Fluid Mech. 433, 135–160.Google Scholar
Srinivasan, S., Wei, Z. & Mahadevan, L. 2017 Wrinkling instability of an inhomogeneously stretched viscous sheet. Phys. Rev. Fluids 2 (7), 074103.Google Scholar
Taylor, G. I. 1969 Instability of jets, threads, and sheets of viscous fluid. In Applied Mechanics, pp. 382–388. Springer.Google Scholar
Tchavdarov, B., Yarin, A. L. & Radev, S. 1993 Buckling of thin liquid jets. J. Fluid Mech. 253, 593–615.Google Scholar
Ward, M. J., Heshaw, W. D. & Keller, J. B. 1993 Summing logarithmic expansions for singularly perturbed eigenvalue problems. SIAM J. Appl. Maths 53 (3), 799–828.Google Scholar